tessellation-based computational methods for the characterization and analysis of heterogeneous...

Composite Science and Technology 57 (1997) 1187-1210 0 1997 Elsevier Science Limited

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TESSELLATION-BASED COMPUTATIONAL METHODS FOR THE CHARACTERIZATION AND ANALYSIS OF

HETEROGENEOUS MICROSTRUCTURES

Somnath Ghosh, Zdzislaw Nowak* & Kyunghoon Lee

Department of Aerospace Engineering, Applied Mechanics and Aviation, The Ohio State University, Columbus, Ohio 43210-1181, USA

(Received 23 July 1996; revised 3 January 1997; accepted 20 January 1997)

Abstract The objective of this paper is to develop a quantitative characterization and response analysis of a multi-phase representative material by the use of Voronoi cells as a unified tool. Voronoi polygons are obtained by Dirichlet tessellation of micrographs, which is the process of discretizing a heterogeneous domain based on the location of a finite set of heterogeneities. Each heterogeneity has associated with it a polygon which is nearer to it than to any other. Composite and porous microstructures with different volume fractions, second- phase counts, shapes, sizes and orientations are generated by computer. The spatial distributions of heterogeneities are random with a given inhibition distance, also known as hard-core distributions. The network of tessellated Voronoi cells is used as elements in the Voronoi cell finite-element method (VCFEM), developed by the authors. Various characterization functions of geometric parameters are generated. VCFEM is executed for plane strain analysis of the microstructures, from which global and local responses are evaluated. Results of the analyses are used for establishing anisotropy measures, and for evaluating statistical functions of stresses. 0 1997 Elsevier Science Limited

Keywords: Voronoi cell finite-element method; quantitative characterization; heterogeneous materials

1 INTRODUCTION

The use of heterogeneous materials with multiple phases dispersed in the microstructure is growing at a rapid rate in modern technological applications. Examples of these materials are metal and alloy systems with precipitates or pores, and composite materials containing dispersions of fibers, whiskers or

* On leave from the Institute of Fundamental Technological Research, Polish Academy of Sciences, 00-049 Warsaw, Swietokrzyska 21, Poland.

particles in the matrix. The overall and local properties of these materials depend on morphological parameters such as size, shape and spatial distribution, and on constituent material properties. A number of micromechanics studies have shown that plastic behavior is highly sensitive to the local configuration because of the non-homogeneous deformation of the ductile constituents. Important studies are by Brockenbrough et al.’ on the effect of fiber distributions, by Christman et a1.2 on the effect of clustering on flow stress and strain hardening, and by Bohm et a1.3 on the flow stress. A thorough understanding of the microstructure-property relationships is a necessary step in effective component design and fabrication, and in the prediction of component behavior and life.

Computational unit-cell models, which generate overall material response through detailed discretization of a representative material element or RME in the microstructure, have been used extensively for the study of various micromechanical aspects of heterogeneous materials2*4*5. To economize computations, a majority of these models make assumptions of local periodicity, thereby implying uniform second-phase distribution. However, for many materials, these simplified unit cells bear little resemblance to the actual stereographic features. Morphological characteristics like size, shape, orientation and spatial distribution of heterogeneities have significant influ- ences on overall macroscopic properties, and in particular on failure properties like ductility, fracture toughness and fatigue for many materials. Studies reflecting the details of actual (not idealized) microstructure are therefore indispensable for establishing microstructure-property relationships.

In recent years, Suresh and co-workers,‘,2.” McHugh et a1.7 and Bohm et a1.3*899 have made novel progress in modeling discontinuously reinforced materials with a random spatial dispersion. In general, large-inclusion aggregates require a very high resolution of finite- element mesh that leads to enormous computational

1187

1188 S. Ghlosh et al.

effort. For some special cases, Bohm et uf.” have constructed special unit cells and boundary conditions to avoid large computations. Ghosh and co-workers have developed a microstructure-based Voronoi cell finite-element model (VCFEM)‘c’2 which can over- come the large computational requirements of conventional finite-element methods by combining the concepts of hybrid finite-element methods with the characteristics of micromechanics. The VCFEM mesh naturally evolves from the microstructure by Dirichlet tessellation to generate a network of multi-sided Voronoi polygons. In VCFEM, each Voronoi cell represents the local microstructural composition, and the analysis needs no further discretization. The effort required to generate a compatible microstructural model is thus drastically reduced. Additionally, computational efficiency is greatly enhanced because the Voronoi cell elements are considerably larger than conventional unit-cell finite elements.

Quantitative characterization of second-phase populations by image-analysis techniques forms an important part of comprehending microstructure- property relationships in heterogeneous materials. Pioneering work on quantitative metallography by the use of tessellation methods was done by Richmond and co-workers.13-‘s. Recently, they have also characterized three-dimensional (3D) distributions simulated from actual two-dimensional (2D) micrographs by pseudo-Saltykov transformation 14and by pair correlation functions.15 Everett and Chu1”*17 used tessellations for determining near-neighbor distances and radial distribution functions for computer-generated patterns. Pyrz18-20 has introduced novel geometric descriptors for stereological quantification and distinguishing between various non-random distributions in heterogeneous microstructures. Everett and Pyrz have mainly concentrated on composites with circular inclusions.

Relatively little work has been done on establishing a link between the overall material response and stereological features. Important contributions in this area are by Brockenbrough et ~21.~~ and by Pyrz.‘87’9 In Ref. 21 an aluminum/silicon composite system was tessellated for characterization and modeled by conventional finite-element analysis. Analytical expressions for stress fields have been derived in Refs 18, 19 for utilization in a unique marked correlation function of stresses.

In this paper, an attempt has been made to establish methods based on Voronoi cells as unified tools for characterizing and modeling multi-phase materials of arbitrary shapes, sizes, orientations and distributions. A similar study has been conducted*’ for understanding the behavior of clustering of identical circular inclusions in composite materials. In the present work, the simulated microstructures differ in volume fraction, second-phase count, shape size and orientation.

Both composite and porous materials are considered in this study. Sixteen different microstructures are generated by computer and tessellated to yield a mesh of Voronoi cells. Various characterization functions based on geometric parameters are developed for comparison of patterns. Elastic-plastic analysis of the microstructures are executed by the Voronoi cell finite-element method in plane strain. Overall stress/ strain relationships and microscopic stress/strain evolution are depicted from these analyses. Measures of geometric and mechanical anisotropy are compared, and finally the statistical functions of different stress measures are evaluated.

2 MICROSTRUCTURE GENERATION AND TESSELLATION

2.1 Computer simulation of heterogeneous microstructures Microstructural representative material elements or RMEs have an important significance, in that they depict a region that is assumed to be representative of the entire microstructure in the neighborhood of a given macroscopic point. A total of 16 RMEs are artificially simulated by computer in order to study the effects of distribution, shape, size and orientation of heterogeneities on geometric and mechanical properties. These are depicted in Fig. 1. Each RME is constructed in a unit (1 X 1) square region with a second-phase volume fraction of 0.10 or 0.20. Two different populations of 25 or 50 heterogeneities are considered. The variation in shape is accommodated through ellipses with different aspect ratios (AR=alb, where a = major axis and b = minor axis of the ellipse). For example, in the case of RMEs with constant shape heterogeneities, the AR is restricted to 1 (circular) or 2.5 only. For microstructures with randomly varying shapes, AR takes arbitrary values between 1 and 2.5. For each volume fraction and second-phase population, the arrangements may be termed: (1) RME-C- uniform circular shape and size; (2) RME-H-uniform elliptical shape and size with horizontal orientation; (3) RME-R-uniform elliptical shape and size with random orientation; and (4) RME-RR-elliptical shape with random aspect ratio and orientation. Heterogeneities are dispersed in hard-core (HC) patterns, which are generated as a variant of a pure random Poisson pattern through the imposition of two constraints. These are: (1) no two heterogeneities are allowed to overlap, and (2) all heterogeneities are completely contained within the RME. These restric- tions are attained by prescribing a minimum permissible distance (MPD) between heterogeneities, and between heterogeneity surfaces and edges of the RME frame. This distance is a function of the second-phase shape and size, and thus of the volume fraction and population. The size of heterogeneities is determined

Characterization and analysis of heterogeneous microstructures 1189

on the basis of the desired volume fraction, V, and the number of heterogeneities, #HET. For random distributions, the range of second-phase area (A,i, and A max in Table 1) as a fraction of RME area, and the range of AR are specified. A random-number generator is used to generate centroidal location and disperse the heterogeneities with a pre-determined value of MPD. The MPD requirement is implemented by incrementing the major and minor axes of each ellipse by the MPD, and verifying intersection with neighboring ellipse. Any event or generation that violates this requirement is discarded. Too many consecutive event rejections lead to alteration of the MPD. Relevant geometrical data used in generating the microstructural realizations are tabulated in Table 1.

W

2.2 Discretization by Diricblet tessellation Dirichlet tessellation is the subdivision of a region, determined by a set of generating points or seeds, such that each seed has associated with it a region that is closer to it than to any other. These subregions are named Voronoi cells. They may be identitled as the basic structural elements of a microstructure corresponding to the basic constitution at this scale. If

P1(xr)rP2(x*) ,. . .,P,(x,,) denote a set of n seeds in a bounded region W, the interior of a Voronoi cell associated with the ith labeled seed Pi is the region Di defined as

D~=[XEWIX-X~I<IX-X~I,V’~#~,P~EW) (1)

The aggregate of all such regions Di constitute the Dirichlet tessellation in plane, as shown in Fig. 1. Each

Fig. 1. Computer-generated microstructures. (a) RME-C and V, = 0.10: (1) #HET = 25, (2) #HET = 50; (b) RME-H and Vf = 0.1: (1) #HET = 25, (2) #HET = 50; (c) RME-R and V, = 0.1: (1) #HET = 25, (2) #HET = 50; (d) RME-RR and V, = 0.1: (1) #HET = 25, (2) #HET = 50; (e) RME-C and Vr = O-2: (1) #HET = 25, (2) #HET = 50; (f) RME-H and Vr = 0.2: (1) #HET = 25, (2) #HET = 50;

(g) RME-R and V,=O.2: (1) #HET=25, (2) #HET=50; (h) RME-RR and V,=O.2: (1) #HET=25, (2) #HET=50.

1190 S. Ghosh et al.

Table 1. Parameters used in the construction of RIMES: volume fraction ( V&, number of heterogeneities (#HET), aspect ratio (u/b), pattern, second-phase major diameter (2a), second-phase minor diameter (2 b), minimum permissible distance between

heterogeneities (MPD), minimum second-phase area (A,,&, maximum second-phase area (A,,)

Vf

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

#HET alb Pattern 2a

25 1.0 RME-C 0.0714 50 1.0 RME-C 0.0504 25 2.5 RME-H 0.1128 50 2.5 RME-H 0.0798 25 2.5 RME-R 0.1128 50 2.5 RME-R 0.0798 25 1.03-2.4 RME-RR 50 1.03-2.5 RME-RR

25 50 25 50 25 50 25 50

I.0 RME-C 0.1010 I.0 RME-C 0.0714 2.5 RME-H 0.1596 2.5 RME-H 0.1128 2.5 RME-R 0.1596 2.5 RME-R 0.1128

1.03-2.4 RME-RR 1.03-2.47 RME-RR

__.______~.

region may be perceived of as the intersection of open half planes bounded by the perpendicular bi-sectors of lines joining the point Pi with each of its neighbors Pi. A two-dimensional mesh generator has been developed for plane sections of multi-phase materials by Ghosh and Mukhopadhyay,” where automatic discretization into basic structural elements is based on the domain boundary, and location, shape and size of the heterogeneities. Effects of the reinforcement shape and size are accounted for by a surface-based tessellation algorithm.

Tessellation of a microstructure into Voronoi cells yields a convenient method for generating geometric descriptors to quantify a given morphology. It naturally identifies regions of immediate influence for each heterogeneity, and also its neighbors corresponding to Voronoi cell edges. This facilitates evaluation of parameters such as the local area fractions, near-neighbor and nearest-neighbor distances and orientations, which are essential in quantitative characterization of the microstructure.

3 QUANTITATIVE CHARACTERIZATION

Microstructural morphology for heterogeneous materials may be appropriately characterized by functions of second-phase shape, size, orientation and spatial distribution. Important studies in quantitative metallography have been conducted by Richmond et al.,”

Pyrz,lR,19 Everett16,17 and Hunt,24 for proposing candidate methods to distinguish between various morphologies. Richmond et al.‘” have considered the distribution of nearest-neighbor distances, and mean distances from near neighbors, in categorizing morphologically different microstructural patterns.

2b MPD ______

0.0714 0.040 0.0504 0.020 0.0452 0,050 0.0320 0.030 0.0452 od40 0.0320 0*020

0.040 0.020

Amin A max

0~00400 0~00200 0~00400 odlO2oo om400 0~00200 oTnIO7o 0.00893 OdlOO36 000395

0.1010 0.010 0.00800 0.0714 0.005 ow400 0.0638 0.010 0QO800 0.0452 oxI oTIO4oo 0.0638 0.010 0~00800 0.0452 0.005 0~00400

0.010 om450 0.01370 0.005 0.00025 0.01278

Everett” has effectively used radial distribution functions, or RDFs, for stereological analysis of distributions with circular heterogeneities. Pyrz’8-20 has also suggested investigations on different distributions by distance-based or cell-related parameters. His studies conclude that the second-order intensity function, K(r), is a very informative descriptor of dispersion. In a comprehensive treatment of damage characterization for aluminum-silicon-magnesium composite systems, Hunt24 has suggested the use of several statistical parameters such as mean, standard deviation and cumulative frequency functions. A number of the geometrical descriptors that have been suggested in these stereological studies are considered in this work to characterize RMEs, as illustrated in Fig. 1.

3.1 Mean and standard deviation of microstructural parameters The statistical mean and standard deviation of local area fraction and surface-to-surface nearest-neighbor distances are computed for each RME, and are tabulated in Table 2. The local area fraction is measured as a ratio of second-phase size to the area of the associated Voronoi cell. Nearest-neighbor distances are determined as minimum distance between surfaces of near-neighbor heterogeneities that share a common Voronoi cell edge. From Table 2, it is clear that the average number of near neighbors (AVNU)

increases with the number of heterogeneities. The average is, however, always less than 6, which is the average number of Voronoi edges for a purely Poisson distribution of points. This may be attributed to facts such as lower second-phase count with distinct edge effects, hard-core as opposed to Poisson distribution,


Table 2. Local statistical measures for characterizing morphology: volume fraction ( V,), number of heterogeneities (#I-ET), aspect ratio (a/b), pattern, mean local area fraction (MAF), standard deviation of load area traction (SDAF), average number of near neighbors (AVNU), mean surface-to-surface newneighbor distance (MSNND), standard deviation of surface-to- surface nemneighbor distance (SDSNND), observed variance of center-to-center near-neighbor distance ( KVND), observed mean of centepto-center nearest-neighbor distance expected mean (MRNND), observed variance of centepto-center nearest-

neighbor distance/expected variance ( VRNND)

V,

Microstructure Statistical parameters

#HET alb Pattern MAF SDAF AVNU MNND SDNND MRNND VRNND

0.1 25 1.0 0.1 50 1.0 0.1 25 2.5 0.1 50 25 0.1 25 2.5 0.1 50 2.5 0.1 25 Ran 0.1 50 Ran

0.2 25 1.0 0.2 50 1.0 0.2 25 2.5 0.2 50 2.5 0.2 25 2.5 0.2 50 2.5 0.2 25 Ran 0.2 50 Ran

RME-C 0.1041 0.0228 4.48 0.0943 0.0312 1.6565 0.3565 RME-C 0.1088 0.0359 4.96 0.0551 0.0245 1.4926 0.4397 RME-H 0.1035 0.0201 4.40 0.0936 0.0215 1.8085 0.0752 RME-H 0.1065 0.0288 4.96 O-0587 0.0174 1.7040 O-0838 RME-R 0.1036 0.0208 4.48 0.0843 0.0223 1.7166 0.0992 RME-R 0.1043 0.0224 4.88 0.0587 0.0125 16411 0.0948

RME-RR 0.1025 0.0504 4.56 0.0858 0.0243 1.5852 0.3995 RME-RR 0.1040 0.0482 5.04 0.0502 0.0188 14615 0.3434

RME-C 0.2085 0.0462 4.48 0.0647 0.0312 1.6565 RME-C 0.2091 0.0448 4.92 0.0395 0.0205 1.5677 RME-H 0.2094 0.0458 4.48 0.0592 0.0201 1.7556 RME-H 0.2067 0.0374 4.92 0.0366 0.0177 1.7542 RME-R 0.2099 0.0468 4.56 oG610 0.0190 1.7748 RME-R 0.2069 0.0357 4.92 0.0374 0.0116 1.7567

RME-RR 0.2161 0.0688 464 0.0543 0.0184 1.5926 RME-RR 0.1789 0.0885 5.04 0.0301 0.0165 1.3606

0.3565 0.3061 O-0181 0.0315 oG425 o-0344 o-1950 0.7309

and surface-based rather than centroid-based tessellation. The mean local area fractions (MAF) are almost always larger than the overall mean area fraction for the RME, with the only exception being for 50 random heterogeneities at V,=O.20. The mean local area fraction does not show much difference for patterns of the same shape and size, and the standard deviation (SDAF) is considerably smaller than the mean. However, the mean changes considerably with randomness in shape, size and orientation, especially at higher area fractions. A relatively large scatter in these area fractions for random patterns is reflected by a sizable increase in standard deviation (SDAF). It can be generally concluded that it is the size rather than the shape which affects the mean and standard of local area fraction.

The mean surface-to-surface nearest-neighbor distance (MNND) decreases with the number of heterogeneities and volume fraction, and is smallest for the RMEs with randomness in size, shape and orientation, at 0.2 area fraction. MNND is generally largest for RME-C and smallest for RME-RR at all volume fractions. Orientation effects on this variable are significantly small in comparison with the shape and size effects, especially at lower volume fractions. Scatter in MNND as seen from the standard deviation (SDNND) is larger for the circular sections. The last two columns in Table 2 correspond to ratios of the mean (MRNND) and variance (VRNND) of the observed center-to-center nearest-neighbor distance to those expected from a pure Poisson point distribution, respectively. The expected mean, E(f), and variance,

E(s2), of nearest-neighbor distance for the Poisson distribution are established in the literature13’16 as:

- 112

where N/A is the area density of points. Richmond et

a1.13 have used these ratios (MRNND and VRNND) to classify actual microstructures as non-random or clustered for small volume fractions ( - O-05). Devia- tions of MRNND and VRNND from unity indicate departure from complete randomness. The microstructures considered here have much larger volume fractions and distinct shapes, and hence the relationships proposed by Richmond et aZ.13 are not quite meaningful. Specifically, MRNND for all volume fractions is found to be greater than unity for all patterns, while VRNND is always less than unity, which seem to correspond with short-range ordering as pointed out previously.13 For RME-RR both MRNND and VRNND are noted to approach unity, indicating stronger bias towards a Poisson distribution.

3.2 Cumulative distribution aud probability density functions The cumulative distribution function, F(x), represents the probability that a random variable X, such as the local area fraction or nearest-neighbor distance, assumes a value smaller than or equal to x. The probability density function, f(x), on the other hand, refers to the probability of a variable X assuming a


value x and may be expressed as f(x) =dF(x)ldx. Pyrz,” Htmtz4 and Richmond et al.‘” have evaluated these functions for nearest- and near-neighbor distances in actual materials. The cumulative distribution and probability density functions of the local area fraction (A) and surface-to-surface nearest-neighbor distance (d) are plotted for all the RMEs in Figs 2 and 3, respectively. The cumulative distribution is normalized with respect to the total number of heterogeneities in the unit square domain.

3.2.1 Local area fraction (A) Figure 2(a,c,e,g) shows the cumulative distribution functions, F(A), whereas Fig. 2(b,d,f,h) shows the density distribution functions, f(A). The peak value of f(A), and thereby the gradients of F(A), progressively diminishes with increasing volume fraction and second-phase populations. For each volume fraction and second-phase population, plots for RME-RR are noticeably different from those of the others. This difference is quite pronounced for low volume fraction with a fewer number of heterogeneities. The high spikes in f(A) for the RME-R at V, = 0.1 [Fig. 2(b)] are consequences of steep gradients in F(A) owing to generally uniform local area fraction. The wide plateaux in f(A) for the random patterns correspond to uniformly increasing area fractions over a larger range, compared with the other RMEs.

3.2.2 Surface-to-surface nearest-neighbor distance (d) Figure 3(a,c,e,g) shows the cumulative distribution functions, F(d), while the density distribution, f(d), is shown in Fig. 3(b,d,f,h). Plateaux in F(d) reflecting zero values in f(d) are especially seen for RME-R, and these correspond to distances for which a nearest neighbor does not exist. Unlike local area fractions, F(d) and f(d) for near-neighbor distances do not clearly distinguish between patterns. Even with increasing second-phase count and volume fraction, there is no distinct trend for the patterns.

3.3 Second-order intensity function and pair distribution function The second-order intensity function, K(r), has been demonstrated to be an informative descriptor of various distributions by Pyrz’*,” because of its sensitivity to local perturbations in otherwise similar distributions. It is defined as the number of additional points or centers of heterogeneities expected to lie within a distance r of an arbitrarily located point, divided by point density. For observations within a finite window W of area A, K(r) corrected for edge effects may be expressed as:18

K(r) = -.$ 2 _!!!$ ,Y k=l %

(2)

where N is the number of points in W. Zk(r) is the number of points in the circle with center at one of the points and radius r, and R, is the ratio of the circumference of a circle of radius r inside W to the entire circumference. For a given r, Zk(r) is calculated for each inclusion by counting the number of additional inclusion centers that lie within a concentric circle of radius r around the inclusion. For circles protruding beyond the RME window, Zk(r) is divided by R, to compensate for the bounded domain. Values of K(r) for various distributions may be compared with that for a pure Poisson distribution of points, known to be m2. The pair distribution function, g(r), on the other hand, corresponds to the probability g(r) d(r) of an finding an additional point within a circle of radius dr and centered at r, where two points are located at r=O and r= r respectively. This is expressed mathematically as:

dK(r) g(r) = +g 7 (3)

Once K(r) is plotted as a function of r, dK(r)ldr is numerically evaluated to obtain g(r). Contrary to K(r), which discriminates between patterns, g(r) quantifies the likelihood of occurrence of near-neighbor distances. For a pure Poisson pattern, g(r) assumes a unit value owing to equal likelihood in occurrence of near- neighbor distances. Figure 4 illustrates K(r) and g(r) plots for microstructural patterns at V,=O*2 as r varies from 0 to approximately one-third of the window size.

The values of K(r) for all patterns are largely lower than those for the Poisson distribution, especially at higher volume fractions, due to the impenetrability constraint. At longer range of r, the K(r) functions for ail patterns converge towards the values for a pure Poisson distribution, suggesting little influence of impenetrability. The g(r) function also approaches unity in an oscillatory manner, but it shows more sensitivity to the particular morphology being considered. The short-range behavior of these functions are, however, different, particularly at lower second- phase population. Figure 4(a) shows a much delayed initiation of K(r) for all patterns in comparison with the Poisson distribution. RME-R exhibits the most delayed response, because of the relatively large minimum distance at which additional heterogeneities may be found. This difference in K(r) from the Poisson distribution diminishes sharply at higher second-phase counts. Local maxima in g(r) plots correspond to most frequent distances, while local minima correspond to the least frequent distances in each pattern. Peaks in g(r) for RME-H and RME-R at low r values are very pronounced, indicating a high frequency of occurrence. These peaks subside with increasing number of heterogeneities. In general, the second-order intensity and pair distribution functions are not as effective in

- RMEC. MiETx.25, Vf=lO%

0.10 0.15 Area fraction

(4

Araa fraction

f 0.9 0.5

.

. - 5 RMEC,WET&5,VF=iV% . E - -- 02 RME-H. #iET=25, vF&O%

--- RME-R,ME~~~,W~XX _

0’ 0.1 ----. . RME-RR. mET-25. VF=20% _

0.0 /‘,‘I I ! 1 0.05 0.10 0.15 020 0.25 0.30 0.35 0.40 0.45

Area fraction

l.or I I 1 I

RMEC, IHETdo, VF=2U?k -- RME-H, #-lETJo, VF=20%

-- - RME=R, OHETa, VF-20%

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 (

Area fraction

Cd

).W

20.0 _

10.0 - ; :,

5.0 I: I

,,_--.

I: ' , .,'

0.0 ! *.

0.00 0.04 0.05 0.12 0.16 0.20 024 Araa fraction

(b) 40.0 r 35.0 t

.b 5 30.0

i p 25.0

- PMEC. MET&O, VF=lO% ---- RME-H.MET&O,VF.l@% - - - RME-R, #HET=&& VF=l 0% ----. RME-RR,#HET=50,VF=lW

% 20.0

f= E 15.0

d 10.0

t

5.0

/-\ : .-I,

o.. ; /JY$__* J

0.00 0.04 0.08 0.12 0.16 020 i 34

Area fraction !4

20.0 r 18.0

- RME-C, #HET=SO, VF=2o% - --. RME-H, #HET-Ki, W&O% - - - RME-R, #HET-50. wQ?O%

RME-RR. #HETSO, VF=ZO%

.’ !

::;I I;.‘, L 0.00 0.05 0.1 0 0.15 0.20 0.25 0.30 0.35 0.40 0

Area fracdon

(4 Fig. 2. Cumulative distribution and probability density functions for local area fraction for: (a) and (b) #HET = 25 and Vf = 0.1;

(c) and (d) #HET = 50 and V, = 0.1; (e) and (f) #HET = 25 and V, = O-2; (g) and (h) #HET = 50 and V, = O-2.

- RME-C, #HETc26, VF=lO%

-.-- RME-H, #HET=25, VF=lU% --- ME-R, #HETz25, VF=lO%

RME-RR, IHET-25, VF=lO%

0.04 0.06 0.06 0.10 0.12 0.14 0.18 0.18 0.20

Nearest neigtxw distance Nearest neigbor distance

04 -1-

- RME-C, XHETSO. VF=lO%

---- RME-H,YHETSO,VF=lO% - -- RME-R. WHET=50, VF=lO%

50.0

45.0

40.0

35.0

30.0

25.0

20.0

15.0

10.0

5.0

0.0 0 .oil

:‘.:,

----- RME-RR, #HET=50, VF=lO%

: : : : ,i’ ‘\

1

,~..,_

1 0.02 0.04 0.08 0.08 0.10 0.12 0

Nearest neigbor distance

1.0 - /-

- RME-C, YHETdO, VF=lO%

- - RME-H. IHETdO, VF=lO%

-- - RME-R, MET&O, VF=lO% RME-RR, #HETdO, VF=lO%

:; 1.14

: .‘,’ 0.0 ’ ” *

0.02 0.04

N%st ne!iEr dis%ke 0.12 (

(4 (4

- RME-C. #HET=25, VFdO%

RME-H, tHET-25, VF&O%

RME-R, #HET=25. VFdXI% RME-RR. #HET=25, VFs?O%

0.0 1 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.18

Nearest neigbor distance

- RMEC. tHET=25, VF=20% - - RME-H, #HET&?5, VF&O%

- - - RME-R, #HET-25, VF&O% RME-RR, WHETc25, VFc2OX

40.0

35.0

0.04 0.06 0.08 0.10 0.12 0

Nearest neigbor distance 0.00 0.02

1.0

0.9

0.6

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0 0.

45.0 1-1 I ’ - RME-C, #HET=50, VF&O%

\ -. - RME-H, #HET=50, VF@O%

.s 40.0 fl, :’ : - - - RME-R. XHETSO, VF=20%

e : ‘, \, RME-RR. #HET=50, Vf=20%

c 35.0 2

: : )

0.02 0.04 0.08 0.08 0.10 Nearest neigbor distance

M

0.00 0.01 0.03 0.07

Nearest ~~gbor%enca 0.06

(h) Fig. 3. Cumulative distribution and probability density functions for surface-to-surface nearest-neighbor distances: (a) and (b) #HET=25 and Vr=O.l; (c) and (d) #HET=50 and V‘=O.l; (e) and (f) #HET=25 and V,=O.2; (g) and (h) #HET= 50 and

v, = 0.2.


distinguishing between various shapes and orientations, as they are for distinguishing between different non-random spatial distributions.”

3.4 Form factor The form factor has been suggested as an effective means of characterizing various shapes of heterogeneity.25 It is defined as

Form factor = 47r X area

perimeter’

where perimeter for elliptical heterogeneities may be approximated as 7r[1*5(a + b) - (ab)“‘] and area is

0.40

0.30

2 0.20 Y

0.10

- RME-C, #HET=25 -.-- RME-H, #HIT=25 - - - RME-R, #HEX25 - -- RME-RR, #HET=25

* Poison

_ .,;., ;.g, ~

0.w L 0.00 0.05 0.10 0.15 020 025 0.30

7.00 r 6.00

5.00

4.00 2 z

3.00

ml

1.00

0.00 0 1.w

(4 ’ @I ’

- RMEC,MET=25 ---- RME-H,#HET=25 --- RME-R, #HIT=25 ----- RME-RR,#HEb25

*Poisson

Tab. This dimensionless factor is unity for uniform circular heterogeneities, and decreases in value with increasing irregularity in shape. The form factor for four RME-RRs at different volume fractions and second-phase numbers [Fig. l(dl,d2,hl,h2)] are plotted as a function of equivalent second-phase diameter in Fig. 5. A linear regression analysis is utilized to express the dependence of form factor on size, and the slope of these lines or regression coefficients are depicted in Fig. 5. The form factor increases significantly with increasing size at lower second-phase count, suggesting that large heterogeneities are more regular than the smaller ones. The regression coeffi- cient decreases with increasing volume fraction,

O.*r__.

c - 0.20 Y

0.10

- RME-C. tHET=25 OJO - ---- RME-H,XHET=25

- - - RME-R, tHEl=25 - --. RME-RR, MET&S

.Pobn

0.00 1 * _I’

. >- ._.‘_--- N-/I- , * ( 0.00 0.05 0.10 0.15 0.20 0.25 0.30

6.00

5.00

4.00

c ‘ij; 3.00

2.00

- RMEG,MET=60 - -. RME-H, #HtT=50 - - - RME-R, XHET=50 ----- RME-RR,#HETdO

aPoisson

0.00 ’ I 0.00 0.05 0.10 0.15 0.20 0.25 0.30

r

Fig. 4. K(r) functions at Vr = 0.2: (a) #HET = 25, (b) #HET = 50; g(r) functions at V, = 0.2: (c) #HET = 25, (d) #HET = 50.


signifying less shape disparity between large and small heterogeneities. This disparity drops sharply for increased number of second phases as is evidenced by the low regression coefficients.

4 MICROMECHANICAL MODELING WITH VCFEM

The Voronoi Cell finite-element model that was successfully developed previously”.” for small- deformation elastic-plastic analysis of heterogeneous materials is briefly presented here. In this model, a

typical RME Y is tessellated into N Voronoi cells occupying a region Y,. The matrix phase of each cell is denoted by Y,,,, the heterogeneity by Y,, and the matrix/heterogeneity interface by aY,. aY, has an outward normal nc and the element boundary aY, has an outward normal ne. An incremental finite-element formulation is developed for rate-independent plastic- ity in terms of an equilibriated stress field or a strain field e (a-load history), compatible displacement fields II and u’ on the element boundary aY, and matrix/inclusion interface aY,, respectively, and a prescribed traction on the traction boundary &,,.

1.00

0.30

8 r; m 0.60 u E b LL 0.40

0.20 I

* RME-RR,#HET=25,VF=10°% - Regression codfiient=0.8730557

0.00 I __I___ L__1- I-_. _i_- _------~i

0.02 0.04 0.06 0.03 0.10

Equivalent Circular Diameter

(a>

* RME-RR,#HETSO,VF=lO% - Regressimcoelfcient=-0.08250172

1.00

0.80

b ii Q 0.60 u E 5 u 0.40

0.20

0.00 0 I.06

1.00

0.80

z 'ii m 0.60 !_L

E z u. 0.40

0.20

* . ‘. - . *

. .

. .

‘. : . *. . .

* RME_RR,tHET=25,VF=20% - Rqresimcxdfiient=O.5593675

-_ 6 _L_._.~, --

0.06 0.10 0.12 Equivalent Circular Diameter

(b)

0.14

. . . . . . ‘: .

‘, 8 .

* . .‘*. , . .’

. . . a’, ***,

. . . ‘* *

, .

. RME-RR,#HETSO,VF=20% - Rqessi0ncxMcient=0.09305033

0.00 L 0.00"""" 4 ” 0.02 0.04 0.06 0.09 0.01 0.03 0.05 0.07 0.09 0.11 0.13

Equivalent Circular Diameter Equivalent Circular Diameter

(c) (4

Fig. 5. Scatter plot of form factor as a function of equivalent diameter for: (a) #HET= 2.5 and V,=O.l; (b) #HET= 25 and V,=O.2; (c) #HET=SO and V,=O.l; (d) #HET=SO and V,=O.2.


Increments in the above quantities are represented by a A preceding the variable. A two-field assumed stress hybrid variational principle is invoked to solve the incremental problem starting from an element energy functional, written as:

l7,,( Acr,Au) = - I

AB(a,Ag) dY - I

e:Aa dY Y, Y,

+ I ((+ + Aa).n’( u + Au) a Y JY,

- I (f + At)-(u + Au) dT r,,

_ c (u”’ + AU” - uc - Auc).nc J ay,

-(u’ + Au’ ) a Y (4) where AB is the increment in element complementary energy. An assumption of a perfectly bonded interface is made through the single-valued displacements u’ on aY,. Superscripts m and c correspond to variables in the matrix and the heterogeneity within each Voronoi cell element. The energy functional for the entire domain is obtained by adding each element functional as 17 = X:1& Independent assumptions on stress increments Au are made in the matrix and the heterogeneity by using stress functions, @(x,y), that result in stress expressions as:

(Au”/“) = [P”““(x,y)]( Apm’c) (5)

where (A/3) are stress coefficients to be determined, and [P] is a matrix of interpolation functions. Element efficiency can be enhanced by choosing stress functions to account for the shape of the heterogeneity near the interface, and also to facilitate interfacial traction reciprocity. Compatible displacement increments on aY, and aY, are generated by interpolation in terms of generalized nodal values as

{ Au~‘~) = [ L”“]( Aqe’c) (6)

where (Aq’} and (Aq’) are generalized displacement increment vectors on aY, and aY,, and [L] is an interpolation matrix. Substituting eqns (5) and (6) into the functional (4), and setting the first variations with respect to Ap” and Apt respectively to zero, results in the following two weak forms of the kinematic relations:

I [PmlT{e + Ae] dY = I

[P”]T[ne][Le]dY(Aqe] Ylll JYe

- f

[P”]T[nc][Lc] dY(Aq”) JYc

I [P”]‘(e + Ae] dY = I

PclTbclWcl d Yi WI (7) Y, JYc

Setting the first variation of 17 with respect to Aq and Aqc to zero results in the weak form of the traction reciprocity conditions as:

I [L”]T[n”]T[Pm]dY a y,

0

- [Lc]T[nc]T[Pm]dY I I

[Lc]T[nc]T[Pc]dY JY, Jy,

For a rate-independent elastic-plastic material following J2 flow theory, the non-linear finite element equations [eqns (7) and (S)] are solved for the stress parameter increments (A/3”, A/3’) and nodal displacements (Aq, Aq”) within an increment. The microscopic Voronoi cell finite-element module is coupled with asymptotic homogenization to yield homogenized elastic-plastic material properties, as discussed else- where.” This procedure assumes a periodic repetition of the microstructural representative material element.

5 STRESS ANALYSIS AND EFFECTIVE PROPERTIES

Numerical simulations are conducted for all 16 RMEs with the multiple-scale finite-element code VCFEM- HOMO. The two-dimensional analyses for evolution of elastic-plastic stress/strain characteristics and overall properties are restricted to plane strain. Two different materials are considered for the same microstructural morphology: (1) an alumina/aluminum composite, where the alumina fiber is assumed to be elastic while the aluminum matrix is elastic-plastic; and (2) a porous aluminum matrix material with voids as second phase. Material properties are as follows.

l Alumina fiber coated with silica:

Young’s modulus (EC) = 344.5 GPa Poisson’s ratio (YJ = 0.26

Aluminum matrix:

Young’s modulus (E,) = 68.3 GPa Poisson ratio (v,) = 0.30 Initial yield stress ( Yo) = 55 MPa Post-yield flow rule: .+ = 0*4841(u,,, - Y,).


5.1 Elastic-plastic analysis of various microstructures Each of the 16 microstructures is subjected to an overall macroscopic strain of 1% in the horizontal direction. The effective stress/strain response resulting from overall straining is illustrated in Fig. 6 for RMEs with V,= 0.2. The elastic portion of the stress/strain response shows very little sensitivity to the micro-

scopic morphology. In this range, the composite is much stiffer than the porous material. For the plastic response, the second-phase count has little effect on the overall response for both materials. However, the effect of distribution on yielding is very pronounced for the porous material in comparison with the composite. For voided materials RME-R is observed

0.10

t

0.08 1

P R 0.08 -

9 .- 8 = 0.04 -

I - - RME-C#HET=25 Q --<1 RME-C,#HETSO - - - RME-H,#HET=25

i

c- - a RME-H,#HET=SO - - - - RME-R,#HET=25

8 - - e RME-R,#HET=50 .. RME-RR,#HET=25

o --o RME-RR,#HET=50

0.00 1 4 -0.001 0.001 0.003 hzotal

Effect %%n 0.009 0.011

0.10 ! - - RME-C#HET=25 Q 4 RME-C,#HET=50 --- RME-H,#HET=PB m - a RME-H,#HET=50

o.08 _ ---- RME-R,#HET=25

z + - - o RME-R,#HETSO

si .---- RME-RR,#HET=25

o -.--..a RME-RR,#Hm+J

0.00 I / I -

-0.001 0.001 0.003 0.005 0.007 0.009 0.011 Effective total strain

0))

-

Fig. 6. Macroscopic effective stress/strain response at 1% strain in the loading direction, for V,= 0.2: (a) composite and (b) porous materials.


to yield at the lowest value of effective stress, while RME-H has the highest yield strength. The overall responses of RME-C and RME-RR are the closest. However, the post-yield behavior, i.e. the effective tangent modulus, is very similar for all distributions in the porous material. For composites, although difference in yielding is not significant, the post-yield tangent moduli vary with distribution. In this case too RME-R and RME-H have the smallest and largest slopes, respectively.

Contour plots of von Mises stress and effective plastic strain at 1% overall strain are depicted for a few representative composites and porous microstructures in Fig. 7 and 8. Figure 7 shows that the largest values of inclusion stresses are reached in RME-H. For random microstructures, RME-RR, bigger inclusions and inclusions in clusters have noticeably higher stresses. A matrix stress histogram for the composite microstructure (not shown) indicates that while the range of stresses is similar for all patterns at the same strain level, their distribution changes with the morphology. For example, the random microstructures exhibit sharper differences in the areas occupied at different stress levels for composites. A different trend is observed for the porous materials, where the differences are more distinct for RME-C and particularly RME-H. The number of heterogeneities only has a slight influence on the stress range. A large extent of matrix yielding at the 1% strain level is evidenced by a large portion of the histograms lying above the stress ratio of unity.

5.2 Anisotropy fiom geometric and response considerations Microstructural morphology generally has a strong effect on directional dependence in the material behavior. Since constituent material properties are assumed to be individually isotropic, a strong correlation is expected between anisotropies arising from geometric dispersion and mechanical response. In this section, two geometric measures for detecting anisotropy are verified against the anisotropic elastic tangent moduli obtained from analysis with VCFEM- HOMO. These descriptors are discussed next.

5.2.1 Angular orientation of heterogeneities Only elliptical heterogeneities with random orientation are subjected to this highly qualitative classification. Circular heterogeneities with hard-core distribution, RME-C, are assumed to be predominantly isotropic, while RME-H is expected to show preferred orientation in the horizontal direction. Histograms depicting orientation of major axes as a function of angle are plotted in Fig. 9 for the range 0” to 180”, with 10” increments. Such histograms have been

discussed as a qualitative measure of degree of orientation.25 The dominance of a single angular orientation due to alignment of heterogeneities in a particular direction is reflected by spikes in the histograms Isotropy is visually inferred if the frequency of occurrence of angular orientations is uniform or random for all subranges, while anisotropy is characterized by a strong preferred orientation. An intuitive classification is presented as ANG in Table 3.

5.2.2 Intercept aspect ratio (MAR = LY,,,,IL”,a A measure of anisotropy defined as the ratio of mean intercept lengths, between Voronoi cell edges, in two orthogonal directions has been introduced pre- viou~ly.‘~ This measure has been slightly altered to accommodate intercept variations at different orientations. Intercept lengths are determined at uniform 10” angular increments by measuring the length of a line between Voronoi cell edges that passes through the inclusion centroid. The intercepts are averaged for all Voronoi cells in the RME. The mean intercept lengths are then plotted as a function of angle (O”-360”). An equivalent ellipse is subsequently constructed by equating the zeroth, first and second moments of the actual plot to those of the ellipse. For each ellipse the ratio of the major axis to minor axis (MAR

= amean&& is then computed; the values are tabulated in Table 3. For an isotropic distribution of inclusions this ratio (MAR) is expected to be unity. The mean difference in MAR from unity is calculated to be 0.1156 with a standard deviation of 0*0&Q. A deviation value of 0.09 from unity (which is lower than the mean) may be intuitively selected as a criterion to differentiate between isotropic and anisotropic patterns. A comparison of ANG and MAR in Table 3 reveals that the two measures concur for a majority of the RMEs.

5.2.3 Effective tangent modulus, E$, The homogenized elastic tangent modulus, E&, is obtained as an orthotropic tensor from the coupled finite-element module VCFEM-HOMO. For both the composite and porous materials, anisotropy parameters A, and A, are evaluated for determining the degree of anisotropy, as tabulated in Table 3. To obtain these, equivalent isotropic components of the elasticity tensor, e.g. Lame constants, are evaluated by equating strain energies for the orthotropic and idealized isotropic materials. The two equations for Lame constants are obtained from (1) biaxial stretching corresponding to a hydrostatic loading state and (2) biaxial tension-compression corresponding to a deviator+ loading state. The anisotropy indices A, = E:JEz and A, = EEJEby’G are ratios of the

S. Ghosh et al.

Fig. 7. Von Mises stress in GPa and effective plastic strain distribution for composite material at 1% overall strain for #HET= 50, V,=O.2: (a) and (b) RME-H; (c) and (d) RME-R; (e) and (f) RME-RR.

Characterization and analysis of heterogeneous microstructures

Fig. 8. Von Mises stress in GPa and effective plastic strain distribution for porous material at 1% overall strain for #HET = 50, V,=O*2: (a) and (b) RME-H; (c) and (d) RME-R; (e) and (f) RME-RR.

Num

ber

of

hete

roge

nelt

les

Num

ber

of h

eter

ogen

eltl

es

‘0

- N

0

L V

I C

D

v O

D

(D

I ,

t

Num

ber

of h

eter

ogen

eiti

es

‘0

- N

0

P

(n

CD

w

0

(0

1 I

Num

ber

of

hete

roQ

enel

Ues

5:

Num

ber

of h

eter

ogen

eiti

es

.a

ru

0

=-

P

6 3 E

--

Num

ber

of h

eter

ogen

eiti

es

.o

N

0 *

VI

S

1

Num

ber

of h

eter

ogen

eltl

es

,o

- N

0

L rn

0

-4

m

(D

S

1


Table 3. Local parameters characterizing overall anisotropy of the microstructure: volume fraction ( V,), patterns, nmber of heterogeneities (WHET), aqlar orientation index (ANG), mean aspect ratio (MAR = a/b), n@sotropy index for composites in XT direction (A xT = E%Ez), anisotropy index for composites in xy direction (A xy = EgJEF’), anisotropy index for porous

material in n direction (A,), anisotropy index for porous material in xy direction (A,)

Microstructure Geometry based Response based

Composite Porous

V, Pattern #HET ANG MAR A, A, A, A,

0.1 RME-C 25 IS0 1.08275 0.99883 1.00062 0.99250 lJ30091 0.1 RME-C 50 IS0 104941 0.99825 lWO53 0.99440 1.00183 0.1 RME-H 25 AN1 1.22560 0.98443 1.01615 0.97538 1.16538 0.1 RME-H 50 AN1 1.20203 O-98267 1.01690 0.97327 1.16335 0.1 RIME-R 25 IS0 1.06728 0.99825 lQO121 0.99668 1.01031 0.1 RME-R 50 AN1 l-10731 099686 0.99738 0.99837 0.98139 0.1 RME-RR 25 IS0 1.03904 0.99891 lW137 0.99860 1.02196 0.1 RME-RR 50 AN1 1.06509 0.99939 1.00080 0.99458 1.00944

0.2 RME-C 25 IS0 1.08285 0.99383 lQO195 0.98353 100832 0.2 RME-C 50 IS0 1.08589 0.99419 099977 O-97382 1JXIO79 0.2 RME-H 25 AN1 1.26342 o-95211 1.04372 0.91862 1.29221 0.2 RME-H 50 AN1 1.29927 0.95364 1.04449 0.93572 1.29551 0.2 RME-R 25 IS0 l-10289 0.99079 0.99578 1.01064 0.95750 0.2 RME-R 50 IS0 1.04247 lQO408 o-99955 0.98993 0.98985 0.2 RME-RR 25 IS0 1.03444 o-99994 1.00630 0.99616 1.04371 0.2 RME-RR 50 AN1 1.09996 0.99736 1@0700 0.97466 1.03290

respective components in the orthotropic tensor to the idealized isotropic tensor. It is observed that the indices for composite and porous materials are slightly different, especially at higher volume fraction. To distinguish between isotropy and anisotropy, a deviation measure DEW = 1/2[ABS(l- A,) + AB- S(l - Axy)] is selected. An arbitrary tolerance can be chosen to create a distinction between anisotropy and isotropy. For example, if DEW 5 0406 for the composite or 5 O-025 for the porous material, then the response may be intuitively classified as isotropic. A comparison of all four measures indicates that the general agreement between all measures is very good. The geometric measures ANG and MAR concur in all but two cases, viz. RME-RR, #HET = 50, V,= O-1 and Rh4E-R, #HET = 25, V, = 0.2. The response measures A, and A, for porous materials agree with one another for almost all patterns except for one, viz. RME-RR, #HET = 50, V,= O-2. In general, most random patterns are found to exhibit predominantly isotropic behavior.

introduced the ‘marked correlation function’ for multi- variate characterization of patterns. A mark may be identified with any field variable (e.g. stresses, plastic strain, etc.) associated with a heterogeneity in the multi-phase domain. The marked correlation function for a heterogeneous domain of area A containing N heterogeneities is mathematically expressed as:19*26

where

M(r) = dH(r)ldr

g(r) ’

H(r) = jj- -j$ 2 i mimk(r) i k=l

(9)

5.3 Marked correlation functions for stresses

where g(r) is the pair distribution defined in Section 3.3 and H(r) is termed as the mark intensity function. A mark associated with the ith heterogeneity is denoted as mi, k’ is the number of heterogeneities which have their centers within a circle of radius r around the ith heterogeneity, for which the mark is mk, and m is the mean of all marks. By definition, M(r) establishes a relationship between the location and associated variables for heterogeneities.

The influence of local morphology on microscopic Marks are identified with four evolving state stress and strain distributions serves as an important variables: (1) maximum principal stress in the inclusion criterion in determining length scales to characterize for composite materials, and (2) maximum principal the representative material element or RME. In this stress, (3) maximum hydrostatic stress and (4) maxi- regard, functions that distinguish between variations in mum von Mises stress in the matrix region, for the stress/strain distributions for local disturbances in Voronoi cell associated with each heterogeneity. microstructural patterns can provide important insight Maximum values associated with each second phase into microstructure-property correlations. Pyrz” has are obtained from data at several sampling points

1204 S. Ghosh et al

within each cell. Plots of M(r) for the various stresses are illustrated in Figs 10-13. In each case, the solid line corresponds to the unit M(r) for regular distributions of circular heterogeneities having identical marks. For arbitrary microstructures, M(r) stabilizes at near-unit values at a distance rinter, at which local morphology ceases to have any significant influence on evolving variables. This distance lintrr is significant in making decisions about length scales, and can provide important information related to the size of the RME.

In general, it is seen that the marks corresponding to the different stress measures yield similar variations in M(r) as a function of r. The highest value of M(r) occurs at minimum r, implying that neighboring

1.6

1.7

.k 1.6 ij c 2

1.5

g 1.4 'G g !

1.3

8 1.2

'0 2 1.1

2 2 1.0

0.9

i.i 'y

0.6 0.0 0.1 0.2 0.3 0.4 0.5 0.6

1.7 -

0.0 0.1 0.2 0.3 0.4 0.5 0.6 r

heterogeneities have the strongest effect on mark intensities. For the composite with 25 inclusions, Fig. 10 shows that M(r) does not stabilize at near-unit values for any of the patterns. Hence the critical influence region rintrr may extend beyond the range of r considered. The matrix hydrostatic stress possesses the strongest mark, especially for RME-R as illustrated by its difference from unity, while marks for the matrix von Mises stresses are weakest. For 50 inclusions in Fig. 11, the plots of stress marks stabilize considerably faster and hence the critical influence region rintzr lies within the window for all but the RME-R microstructure. This influence region is quite small for RME-RR, indicating smaller representative material elements. A slightly different behavior is

l.er', I ', 1 1.7

,k 1.6 - ii = 2

1.5

g 1.4; ._

z 1.3

b 1.2

*-d Principal Stressinmattix d ~PdndpalStressinindusion +--+VanMiasSlraesinmahix *--Hyd&a!kSlmssinmatti -u&m

0.8' ' ' 0.0 0.1 0.2 0.3 0.4 0.5 0.6

r

,g 1.6 ti c .z

1.5

g 1.4 'G j 1.3

1 1.2

0 8 1.1

2 2 1.0

0.9

*-dPrindpalStressin matrix Q. 4PdndpalStfessininchkn +--+VonMsesSlmssinm it-otiykWItllUcSIm611innWix -uniirm

0.6 J 0.0 0.1 0.2 0.3 0.4 0.5 0.6

r

Fig. 10. Marked correlation functions, M(r), in composite microstructure for #HET=25 and Vf= 0.2: (a) RME-C; (b) RME-H; (c) RME-R; (d) RME-RR.


noted for marks in the porous microstructures in Figs 12 and 13. For 25 voids, plot of the microstructures RME-C and RME-H stabilize rather quickly, while the other two do not stabilize within the range of r considered. However, a much faster decay is observed for RME-R and RME-RR, as the void count increases to 50. In summary, M(r) decays rapidly with increasing heterogeneity count, and for it does not stabilize easily for random orientations with the same shapes and sizes.

5.4 Cumulative distribution functions for stresses Figures 14 and 15 show the cumulative distribution functions of three different normalized stress measures, i.e. (1) maximum inclusion principal stress, (2) maximum matrix von Mises stress and (3) maximum

.$ 1.6

5 c 2 1.5

5 1.4 'S t 1.3

1 1.2

f 1.1

g 1.0

0.9 :

0.6 4 a 0.0 0.1 0.2 0.3 0.4 0.5 0.6

I=

r

.:

A---bprincpalstreghmatfill

t---ai'iinc@dStmhhvzh&m +--+VmMn8sllwsinr

~-9HydmtaGcslmaninm

-IJdfUlM

0.0 0.1 02 0.3 0.4 0.5 0.6 r

(c)

matrix hydrostatic stress, for the composite and porous materials. The maximum corresponds to the highest value within each Voronoi cell. These stress measures are then normalized with the averaged corresponding stress measure for the entire microstructure. In the composite, the distribution function for RME-H is distinctly different from the others, especially for inclusion principal stress and matrix hydrostatic stress. The plot for principal stresses in the inclusions shows the maximum difference. This indicates that generally higher values of these stresses are achieved in this microstructure. The larger plateau in the distribution functions at higher volume fraction corresponds to large dispersions in stress values. For the porous material, however, the distribution functions for RME- R with random orientation differ significantly from the

1.6

1.7

.kj 1.6 5 c 2

1.5

s 1.4 'S 3 : 1.3

8 1.2

-J 1.1

2 1.0

0.9

1.6

.E 1.6 5 c: 2

1.5

5 1.4 ._

5 1.3

1 1.2

u Q) iz

1.1

g 1.0

0.9

0.6 D.0

P

0.1 02 0.3 0.4 0.5 0.6

I-I

0.1 0.2 0.3 0.4 0.5 0.6 r

(4

Fig. 11. Marked correlation functions, M(r), in composite microstructure for #HET=50 and V,=O-2: (a) RME-C; (b) RME-H; (c) RME-R; (d) RME-RR.


others, particularly at the lower void count. At high void population, the difference between the patterns increases.

6 DISCUSSIONS AND CONCLUSIONS

The aim of this paper was to establish the usefulness of Dirichlet tessellation of heterogeneous domains for quantitative metallography and response modeling. Unified tools of characterization based on Voronoi cells and finite-element modeling are implemented to link these two important areas of micromechanics. In particular, computer-simulated hard-core microstructures with variations in shapes, sizes and orientations are investigated for correlations between functions representing geometry and mechanical response. Both

1.7 -I

,E 1.6 b-dPftnc#atStresinmatrk ij +-+VonMiwsStmsinmatti c 2 1.5 o--atty&mtaticSlrs8inmatitx

-UnttOml g 1.4 'Z g L

1.3

$ 1.2

0 3 1.1

kJ 5 1.0

0.9

i.a

1.7

j 1.6 ij c 2

1.5

0' 1.4 ._

4 !!

1.3

8 1.2

u 3 1.1

2 2 1.0

0.9

‘r

0.2 0.3 0.4 0.5 0.6

r

(a)

+-dPiindpaJ.Stresainmati t-+VcnkeoSlressinmatrix o--ottfddaikStressinmatioc - Unwm

P

-

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composite materials with inclusions and porous materials with voids are considered in this study. A total of 16 different microstructures are simulated by computer to represent different numbers, sizes, shapes, orientations and volume fractions of heterogeneities.

In the first part of this paper, Voronoi cells are utilized to obtain stereologic information for the different morphologies. Geometric parameters such as local area fraction, surface-to-surface nearest-neighbor distance, shape deviation from a perfect circle and angular orientation are assumed to characterize the microstructural composition adequately. Various statistical functions such as mean and standard deviations, cumulative distribution and probability density functions, second-order intensity and pair distribution functions of the above parameters are then examined

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for qualitative and quantitative distinction between the patterns. While overall volume fraction generally has a very strong influence, other variables like number, shape and size, also affect the characterization functions considerably. For example, the mean local area fraction is sensitive to shape, size and orientation, while the mean nearest-neighbor distance is less sensitive to orientation. The cumulative distribution and probability density functions for local area fractions, F(A) and f(A), distinguish between patterns, whereas those for the nearest-neighbor distance, F(d) and f(d), do not. Also the intensity functions, K(r) and g(r), are not as effective for these patterns as they are for non-random spatial distributions in Ref. 18. Form factor is observed to be a good indicator of shape variation in different patterns.

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Elastic-plastic stress analysis is conducted with the Voronoi cell finite-element method to evaluate macroscopic properties and local evolution. Macro- scopic stress/strain plots lead to different inferences for composite and porous materials. While yield points are relatively unaffected by the microstructural pattern for the composite, they vary considerably with morphology for the porous materials. On the other hand, post-yielding behavior and effective tangent modulus are much more sensitive to the pattern for the composite than for the porous material. Geometric and mechanical anisotropy are gaged by estimators based on orientation and tangent modulus. For a qualitative evaluation of deviation from isotropy, arbitrary small tolerances are chosen. This study shows that relatively good agreement is obtained between

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the anisotropy descriptors. As is expected, random patterns exhibit higher levels of isotropy.

To discuss the influence of local morphology on stress manifestations, marked correlation functions of various stress measures are considered. This function for von Mises stress in the matrix is found to be least

affected by dispersion patterns while it is quite sensitive to principal stresses and matrix hydrostatic stress. The response is quite different for composites and porous materials owing to the difference in the non-linear plastic zones. For composites, RME-H shows a greater difference in M(r) from the others,

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( f 1 Fig. 14. Cumulative distribution functions for various stresses in the composite microstructure for #HET = 25 and 50: (a) and (d) maximum normalized principal stress in inclusion; (b) and (e) maximum normalized von Mises stress in matrix; (c) and (f)

maximum normalized hydrostatic stress in matrix.


while for porous materials it is the RME-R that is at variance with others. Finally, the cumulative distribution functions for stresses are plotted and similar trends as with M(r) are noted. In conclusion, characterization and modeling tools based on Voronoi cells are found to be very effective in the study of arbitrary microstructures. Observations made in this

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study may be used with confidence in the characterization and modeling of actual heterogeneous materials, which will be presented in a future paper.

ACKNOWLEDGEMENTS

This work has been sponsored by the Mechanics and Materials program (Program Director: Dr 0. Dillon)

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(e) Fig. 15. Cumulative distribution functions for various stresses in the porous microstructure for #HET=25 and 50: (a) and (c)

maximum normalized von Mises stress in matrix; (b) and (d) maximum normalized hydrostatic stress in matrix.


of the National Science Foundation through grant no. MSS-9301807, by the United States Army Research Office through grant no. DAAHO4-95-1-0176 (Pro- gram Director: Dr K. R. Iyer), and by a grant from the ALCOA Technical Center. Computer support by the Ohio Supercomputer Center through grant #PAS813-2 is also gratefully acknowledged.

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